How did the European participants of the First Crusade deal with the language barrier between themselves and the locals?

The first thing to note is that medieval society was used to overcoming language barriers. Today, the people of France predominantly only speak one language - modern French - but in 1100 they spoke several, with the big divide being between Old French in the north (which in turn had many dialects that could be significantly different from each other) and Occitan in the south (which, again, many dialects). That's to say nothing of the Flemish speakers in the north-east corner of France, the Bretons in Brittany, or German speakers on the border between the kingdom of France and the Holy Roman Empire. Medieval Europe was a linguistic melting pot, so to communicate between different groups required translators. Translators were so common that most sources won't even mention their presence.

The most widely understood language was Latin. It was the language of the church, so anyone who received training from the church knew how to read and write in it. Generally speaking, translation in the Middle Ages looked like this:

Local language 1 > Latin > Local language 2

Given that crusades were drawn together from a variety of linguistic backgrounds, the use of Latin as a go-between was extremely common. Often, sermons would be given in Latin and then translated into local languages for different groups. This worked well for the crusaders to talk amongst themselves.

But the local languages of the Middle East were Arabic and Greek, which in turn could have many local dialects. Then for diplomacy you'd have to deal with Turkish warlords who spoke Turkish but perhaps did not speak Greek or Latin. To overcome this, the crusaders employed traders or pilgrims who had been to the region before and might have known some of the language. On the First Crusade, the Byzantine emperor sent the general Tatikios, a Christian Turk, to assist them. The crusaders also had a man named Hurluin who acted as an envoy between the crusaders and the Turkish warlord Kerbogha, though we know almost nothing about him beyond his utility as a translator. His name is of Norman origin, so it is possible that he was an Italian Norman who had interacted with the Muslim communities of Sicily and learned some Arabic that way, but we don't know. The Normans of southern Italy who ruled over the Muslims of Sicily regularly put documents into Norman, Latin and Arabic so they may have brought translators with them. They may also have employed local translators, but again we don't know much about it because translators were so common that writers didn't think to mention them often.

They may also have used phrasebooks; one surviving example from the late 11th century contains phrases useful for pilgrims and traders working from Latin into Greek, for example. Learning stock phrases like this would have helped them in basic interactions, like buying food at local markets, which would have been a logistical challenge very early on. Contemporary sources write of the crusaders simply going around markets buying things, so they must have at least learned the basic phrases in order to do this without requiring translators at every market stall. Later on as some crusaders settled in the Holy Land, we know they made a concerted effort to learn local languages, especially Arabic, and some may have begun to do so during the crusade itself.

So we're not entirely sure, but the limited evidence we have suggests that the Byzantines provided some interpreters, the southern Italian Normans probably brought their own, many learned basic phrases to interact with locals, and some who planned to settle may have begun learning Arabic as soon as they could.